AoPS - Problem

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Problem

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Tags: geometry, circumcircle, geometry unsolved

RaleD

08.04.2011 23:06

Let $H$ be orthocenter and $O$ circumcenter of an acuted angled triangle $ABC$ . $D$ and $E$ are feets of perpendiculars from $A$ and $B$ on $BC$ and $AC$ respectively. Let $OD$ and $OE$ intersect $BE$ and $AD$ in $K$ and $L$ , respectively. Let $X$ be intersection of circumcircles of $HKD$ and $HLE$ different than $H$ , and $M$ is midpoint of $AB$ . Prove that $K, L, M$ are collinear iff $X$ is circumcenter of $EOD$ .

mahanmath

08.04.2011 23:34

I think there is a typo in problem ,I`ve found that the circumcenter of $EOD$ always lies on $OX$ and that its never coincide with $X$ . Sorry if I`m an idiot !

Vo Duc Dien

12.04.2011 22:16

There's absolutely something wrong with the problem. K, M and L are never concurrent since angles MAL and MBK are less than 180 degrees.

Vo Duc Dien

12.04.2011 22:29

If you can read Serbian language or whatever it is, their official problem and solution link is http://srb.imomath.com/zadaci/2011_smo_resenja.pdf Good luck!

Vo Duc Dien

12.04.2011 22:37

The graphic is a little different and the problem wording is actually correct as I have seen their solution by just looking at their graphic for this problem on that web link. Sorry!

Vo Duc Dien

13.04.2011 06:48

The problem just does not apply to any configuration. That's all. Weird! Too busy to find out.

DC93

19.06.2011 05:44

I have a solution in Chinese,but I haven't enough time to translate it into English See here: http://bbs.eduu.com/thread-792038-1-1.html 7#~12#

RaleD

15.01.2012 20:28

$E, D, H, O$ are concyclic